Discrete Mathematics with Graph Theory (3rd Edition) 125

Discrete Mathematics with Graph Theory (3rd Edition) 125 -...

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Section 5.1 123 (d) rng f = {n E Z I n ~ 998}. 26. [BB] (a) i. 0 E nZ because 0 = O(n) is a multiple of n. ii. If a E nZ, then a = kn for some k, so -a = ( -k)n E nZ because it is also a multiple of n. iii. If a, bEnZ, then a = kIn and b = k2n for some integers kl' k2' so a+ b = (kl +k2)n E nZ because it is also a multiple of n. (b) i. 0 E A by definition of ideal, so if A contains just one element, that element must be 0, in which case A = OZ is of the form nZ. ii. If A contains more than element, it contains a nonzero element a. By definition of ideal, A contains both a and -a, one of which is positive. iii. The set of positive integers in A is not empty (by ii.) and, hence, contains a smallest element by the Well-Ordering Principle (4.1.2). iv. First, In = n E A. Then, if kn E A for some k > 0, so also is (k + l)n E A, by definition, because (k + l)n = kn + n. By the Principle of Mathematical Induction, we conclude that kn E A for all k > O. Since (-k)n = -kn is the negative of kn,
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.

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